|© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology:||(34)||
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Solve 5 of the following 6 problems. Each problem is worth 20 points. If you solve more than 5 problems indicate very clearly which ones you want graded; otherwise a random one will be left out at grading and it may be your best one! You have an hour and 50 minutes. No outside material other than stationary is allowed.
Problem 1. Let be an arbitrary topological space. Show that the diagonal , taken with the topology induced from , is homeomorphic to . (18 points for any correct solution. 20 points for a correct solution that does not mention the words ``inverse image'', ``open set'', ``closed set'' and/or ``neighborhood''.)
Problem 2. Let be a connected metric space and let and be two different points of .
Problem 5. The ``diameter'' of a metric space is defined to be .
Problem 6. If is a sequence of continuous functions such that for each , show that is continuous at uncountably many points of .